One-Port Network Theory

The basic idea of a one-port network [524] is shown in
Fig. 7.5. The one-port is a ``black box'' with a
single pair of input/output terminals, referred to as a port. A
force is applied at the terminals and a velocity ``flows'' in the
direction shown. The admittance ``seen'' at the port is called the
driving point admittance. Network theory is normally described
in terms of circuit theory elements, in which case a voltage is
applied at the terminals and a current flows as shown. However, in
our context, mechanical elements are preferable.

Figure 7.5:
A one-port network characterized
by its driving point admittance . For any applied force
, the observed velocity is
.

In a physical situation, if two elements are connected in such a way
that they share a common velocity, then they are in series. An example
is a mass connected to one end of a spring, where the other end is attached
to a rigid support, and the force is applied to the mass, as shown in
Fig. 7.7.

When two physical elements are driven by a common force (yet
have independent velocities, as we'll soon see is quite possible),
they are formally in parallel. An example is a mass connected
to a spring in which the driving force is applied to one end of the
spring, and the mass is attached to the other end, as shown in
Fig.7.11. The compression force on the spring
is equal at all times to the rightward force on the mass. However,
the spring compression velocity does not always equal the
mass velocity . We do have that the sum of the mass velocity
and spring compression velocity gives the velocity of the driving point,
i.e.,
. Thus, in a parallel connection, forces
are equal and velocities sum.

Thus, the impulse response of the mass oscillates sinusoidally with
radian frequency
, and amplitude . The
velocity starts out maximum at time , which makes physical sense.
Also, the momentum transferred to the mass at time 0 is
;
this is also expected physically because the time-integral of the applied
force is 1 (the area under any impulse is 1).

In any mechanical situation we have , in principle, since at
sufficiently high frequencies, every mechanical system must ``look like
a mass.''8.3 However,
for purposes of
approximation to a real physical system, it may well be best to
allow and consider the above expression to be a
rational approximation to the true admittance function.

Figure 7.14:Poles and zeros of a lossless
immittance (reactance or suseptance) must
interlace along the Axis. Left: Pole-zero plot.
Right: Phase response. The ``spring/mass'' labels along the
frequency axis correspond to the case of a lossless
admittance (susceptance) in which a spring admittance
(
) gives a phase shift, while
that of a mass (
) gives a
phase shift between the input driving-force and output velocity.

Referring to Fig.7.14, consider the graphical method for
computing phase response of a reactance from the pole zero diagram
[449].8.4Each zero on the positive axis contributes a net 90 degrees
of phase at frequencies above the zero. As frequency crosses the zero
going up, there is a switch from to degrees. For each
pole, the phase contribution switches from to degrees as
it is passed going up in frequency. In order to keep phase in
, it is clear that the poles and zeros must strictly
alternate. Moreover, all poles and zeros must be simple, since a
multiple poles or zero would swing the phase by more than
degrees, and the reactance could not be positive real.

The positive real property is fundamental to passive immittances and
comes up often in the study of measured resonant systems. A practical
modeling example (passive digital modeling of a guitar bridge) is
discussed in §9.2.1.